## Commutativity for a certain class of rings.(English)Zbl 0839.16029

For rings $$R$$, the author considers four similar conditions involving commutators, of which the first is $$(c_1)$$: “For every $$x, y\in R$$ there holds $$x^r[x^s, y]= \pm[ x, y^t] x^n$$ with integers $$t> 1$$, $$s\geq 1$$, $$n\geq 0$$, $$r\geq 0$$.” His theorems assert that a ring $$R$$ with 1 must be commutative if it satisfies one of these conditions in conjunction with an appropriate restriction on additive torsion. Apparently $$t$$, $$s$$, $$n$$, and $$r$$ are fixed, but it is not clear how to interpret the ambiguity in sign in condition $$(c_1)$$ and the other three conditions. The safe interpretation is that one chooses the same sign for all $$x, y\in R$$.
While the theorems of this paper are similar to many known results, the proofs do have an element of novelty; they provide an interesting application of some earlier results of T. P. Kezlan [Math. Jap. 29, 135-139 (1984; Zbl 0538.16028)].

### MSC:

 16U70 Center, normalizer (invariant elements) (associative rings and algebras) 16U80 Generalizations of commutativity (associative rings and algebras) 16R50 Other kinds of identities (generalized polynomial, rational, involution)

### Keywords:

commutativity of rings; commutator constraints

Zbl 0538.16028
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### References:

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